Embedded newform 7.17.d.a.3.7

• Label: 7.17.d.a.3.7
• Level: $7$
• Weight: $17$
• Character: 7.3
• Analytic conductor: $11.363$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	7.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3627180700\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^19 + 470221*x^18 - 17792382*x^17 + 149610140535*x^16 - 5885709493452*x^15 + 25413927970368382*x^14 - 921982218074544860*x^13 + 3092605513604061172669*x^12 - 78217760507911051825254*x^11 + 217775891572550635786539387*x^10 - 772894989756975230685011586*x^9 + 10848813785145958185022965942709*x^8 + 50328102663961537370536178622316*x^7 + 292280160701254761307491650568975646*x^6 + 6884264619587692201297746565224842172*x^5 + 5465648386210730671112398026225318665199*x^4 + 69498817213798305204024309903431721187110*x^3 + 44558299163904360256389313185385225104835245*x^2 - 34923314543744821707168835219945993579648950*x + 261490399505792227817024003348104130194641159225
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{38}\cdot 3^{16}\cdot 7^{22} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3.7
Root			\(90.3350 - 156.465i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.3
Dual form			7.17.d.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(94.8350 + 164.259i) q^{2} +(-9421.11 - 5439.28i) q^{3} +(14780.6 - 25600.8i) q^{4} +(-477118. + 275464. i) q^{5} -2.06334e6i q^{6} +(5.57858e6 + 1.45338e6i) q^{7} +1.80371e7 q^{8} +(3.76482e7 + 6.52085e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	94.8350	+	164.259i	0.370449	+	0.641637i	0.989635	−	0.143608i	\(-0.0458704\pi\)
							−0.619185	+	0.785245i	\(0.712537\pi\)
\(3\)	−9421.11	−	5439.28i	−1.43593	−	0.829032i	−0.438362	−	0.898798i	\(-0.644441\pi\)
							−0.997563	+	0.0697663i	\(0.977775\pi\)
\(5\)	−477118.	+	275464.i	−1.22142	+	0.705188i	−0.965221	−	0.261435i	\(-0.915804\pi\)
							−0.256201	+	0.966624i	\(0.582471\pi\)
\(7\)	5.57858e6	+	1.45338e6i	0.967698	+	0.252113i
							\(11\)	7.76109e7	−	1.34426e8i	0.362060	−	0.627107i	−0.626239	−	0.779631i	\(-0.715407\pi\)
0.988300	+	0.152524i	\(0.0487401\pi\)
\(13\)			1.23204e9i			1.51036i	0.655519	+	0.755179i	\(0.272450\pi\)
							−0.655519	+	0.755179i	\(0.727550\pi\)
\(17\)	−1.42412e9	−	8.22217e8i	−0.204153	−	0.117868i	0.394438	−	0.918923i	\(-0.370939\pi\)
							−0.598591	+	0.801055i	\(0.704273\pi\)
\(19\)	1.62597e10	−	9.38754e9i	0.957379	−	0.552743i	0.0620134	−	0.998075i	\(-0.480248\pi\)
							0.895365	+	0.445332i	\(0.146915\pi\)
\(23\)	3.57099e10	+	6.18513e10i	0.456001	+	0.789817i	0.998745	−	0.0500813i	\(-0.0159481\pi\)
							−0.542744	+	0.839898i	\(0.682615\pi\)
\(29\)	−1.65746e11			−0.331329			−0.165664	−	0.986182i	\(-0.552977\pi\)
							−0.165664	+	0.986182i	\(0.552977\pi\)
\(31\)	7.81270e11	+	4.51066e11i	0.916025	+	0.528868i	0.882365	−	0.470566i	\(-0.155950\pi\)
							0.0336605	+	0.999433i	\(0.489283\pi\)
\(37\)	1.77083e12	+	3.06716e12i	0.504153	+	0.873219i	0.999988	+	0.00480224i	\(0.00152861\pi\)
							−0.495835	+	0.868417i	\(0.665138\pi\)
\(41\)			7.99933e12i			1.00180i	0.865504	+	0.500902i	\(0.166998\pi\)
							−0.865504	+	0.500902i	\(0.833002\pi\)
\(43\)	−8.81832e11			−0.0754464			−0.0377232	−	0.999288i	\(-0.512011\pi\)
							−0.0377232	+	0.999288i	\(0.512011\pi\)
\(47\)	−3.12713e12	+	1.80545e12i	−0.131330	+	0.0758232i	−0.564225	−	0.825621i	\(-0.690825\pi\)
							0.432896	+	0.901444i	\(0.357492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.17.d.a.3.7	✓	20
7.3	odd	6		49.17.b.a.48.7		20
7.4	even	3		49.17.b.a.48.8		20
7.5	odd	6	inner	7.17.d.a.5.7	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.17.d.a.3.7	✓	20	1.1	even	1	trivial
7.17.d.a.5.7	yes	20	7.5	odd	6	inner
49.17.b.a.48.7		20	7.3	odd	6
49.17.b.a.48.8		20	7.4	even	3